Relaxation of second order geometric integrals and non-local effects

Abstract

We are concerned with the relaxation of second-order geometric integrals, i.e., functionals of the type:C c ∞ (ℝ N )∋u↦F μ (u):=∫ ℝ N f∇ 2 u (x)dμ(x),where ∇ 2 u is the Hessian of u, f:MsymN→[0,+∞] is a continuous function, and μ is a finite positive Radon measure on ℝ N . A relaxation problem of this type was studied for the first time by G. Bouchitté and I. Fragala , where they pointed out a new phenomenon: the functional relaxed of F μ has, in general, a 'non-local' representation. Working on a more formal level than in, we develop an alternative method making clear this 'strange phenomenon'.

Abstract

We are concerned with the relaxation of second-order geometric integrals, i.e., functionals of the type:C c ∞ (ℝ N )∋u↦F μ (u):=∫ ℝ N f∇ 2 u (x)dμ(x),where ∇ 2 u is the Hessian of u, f:MsymN→[0,+∞] is a continuous function, and μ is a finite positive Radon measure on ℝ N . A relaxation problem of this type was studied for the first time by G. Bouchitté and I. Fragala , where they pointed out a new phenomenon: the functional relaxed of F μ has, in general, a 'non-local' representation. Working on a more formal level than in, we develop an alternative method making clear this 'strange phenomenon'.

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